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Game Theory

Game Theory. Topic 8 Auctions. “Everything is worth what its purchaser will pay for it.”. - Publilius Syrus (Maxim 847, 42 B.C.). What is an Auction?. Definition: A market institution with rules governing resource allocation on the basis of bids from participants

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Game Theory

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  1. Game Theory Topic 8Auctions “Everything is worth what its purchaser will pay for it.” - Publilius Syrus (Maxim 847, 42 B.C.)

  2. What is an Auction? • Definition: • A market institution with rules governing resource allocation on the basis of bids from participants • Over 30% of US GDP moves through auctions: • IPOs • Emissions permits • Radio Spectrum • Import quotas • Mineral rights • Procurement • Wine • Art • Flowers • Fish • Electric power • Treasury bills

  3. Sample Auction “Mistakes are the portals of discovery” - James Joyce

  4. Going Once, Going Twice, … Bidding starts at $1 Who will make the first bid?

  5. Overview of Auctions • Auctions are a tricky business • Different auction mechanisms • sealed vs. open auctions • first vs. second price • optimal bidding & care in design • Different sources of uncertainty • private vs. common value auctions • the winner’s curse

  6. Private Value Auction • Dinner

  7. Common Value Auction • Unproven oil fields

  8. Sources of Uncertainty • Private Value Auction • Each bidder knows his or her value for the object • Bidders differ in their values for the object • e.g., memorabilia, consumption items • Common Value Auction • The item has a single though unknown value • Bidders differ in their estimates of the true value • e.g., FCC spectrum, drilling, disciplinary corporate takeovers

  9. Basic Auction Types • Open Auctions (sequential) • English Auctions • Dutch Auctions • Japanese Auctions • Sealed Auctions (simultaneous) • First Price Sealed Bid • Second Price Sealed Bid

  10. English Auctions (Ascending Bid) • Bidders call out prices (outcry) • Auctioneer calls out prices (silent) • Bidders hold down button (Japanese) • Highest bidder gets the object • Pays a bit over the next highest bid

  11. Dutch (Tulip) Auction Descending Bid • “Price Clock” ticks down the price • First bidder to “buzz in” and stop the clock is the winner • Pays price on clock

  12. Sample Dutch AuctionMinimum Bid: $10

  13. WINNER! Pays $700 $700 $400 $500 $300 Sealed-Bid First Price Auctions • All buyers submit bids • Buyer submitting the highest bid wins and pays the price he or she bid

  14. WINNER! Pays $500 $700 $400 $500 $300 Sealed-Bid Second Price Auctions • All buyers submit bids • Buyer submitting the highest bid wins and pays the secondhighest bid

  15. $500 $400 $300 Why Second Price? • It is strategically equivalent to an English Auction

  16. Why Second Price? • Bidding strategy is easy • Bidding one’s true valuation is a (weakly) dominant strategy • Intuition: • The amount a bidder pays is not dependent on her bid

  17. Bidding True Valuation Say your value is $100 • Why not bid $500? • If others all bid under $100, no difference • If someone bids > $500, no difference • If someone bids $300, you overpay! • Why not bid $50? • If someone bids $80, you lose (but would have made money bidding $100)

  18. First Price Auction • First price auction presents tradeoffs • If bidding your valuation–no surplus • Lower your bid below your valuation • Smaller chance of winning, lower price • Bid shading • Depends on the number of bidders • Depends on your information • Optimal bidding strategy is complicated!

  19. Which is Better? • In a second price auction • bidders bid their true value • auctioneer receives the second highest bid • In a first priceauction • bidders bid below their true value • auctioneer receives the highest bid

  20. Revenue Equivalence • All common auction formats yield the same expected revenue (in theory) Any auctions in which: • The prize always goes to the person with the highest valuation • A bidder with the lowest possible valuation expects zero surplus yield the same expected revenue

  21. Revenue Equivalence in the Real World • Risk Aversion • Does not influence 2nd price auctions • Risk averse bidders are more aggressive in first price auctions • Risk aversion  1st price or Dutch are better • Non-familiarity with auctions • More overbidding in second-price auctions • More overbidding in sealed-bid auctions • Inexperience  2nd price sealed bid is better

  22. Designing Auction Rules • Every rule may have unintended consequences • What is the minimum bid for a new bidder? • How much must bids be beaten by?

  23. Importance of RuleseBay … • Three laptops for sale • Top three bidders pay the third highest bid • Opening bid: $1 • Current high bids: • $50, $80, $400 • How high should the next bid be?

  24. Importance of Rules FCC Spectrum Auctions… • Discouraging Collusion • Do not identify highest bidders • Capturing Surplus • Do not set a bidding increment “I bid $8,000,483” “I bid $3,000,395”

  25. Summary • Bidding: • Bid true valuation in 2nd price auctions • Shade bids in 1st price auctions • Designing: • Take advantage of inexperience • Take advantage of risk aversion • Do sweat the little stuff

  26. Sources of Uncertainty • Private Value Auction • Difficult to lose money • Do not bid more than your value (or less than your cost) • Common Value Auction • The item has a single though unknown value • Bidders differ in their estimates • The winner might be wrong!

  27. Common Value Auctions • Example: Offshore oil leases • Value of oil is roughly the same for every participant • No bidder knows value for sure • Each bidder has some information • Auction formats are not equivalent • Oral auctions provide information • Sealed-bid auctions do not

  28. Bidder 2 Bidder 1 Bidder 4 Bidder 3 Hypothetical Oil Field Auction 5 4 3 2 1 10 9 8 7 6 10 tracts for sale each with four bidders

  29. Bidder 2 Bidder 1 Bidder 4 Bidder 3 Hypothetical Oil Field Auction • Each tract has four bidders • Each bidder knows the amount of oil in his or her quadrant • Each quarter’s value is evenly distributed between $200,000 and $800,000 • Total value of oil field: Sum of the values of the four quarters • Type of auction: First price sealed bid

  30. Oil Field Auction • How much do you bid?

  31. The Winner’s Curse $40 • The estimates are correct, on average What happens if everyone bids his or her estimate? $70 $50 $60 $80 $60

  32. The Winner’s Curse Defined • If the average estimate is generally correct, the highest estimate is usually too high • If bids are based on estimates, the highest bidder overpays • To avoid the winner’s curse, estimate the average of the object conditional on winning the auction

  33. Avoiding the Winner’s Curse • Given that I win an auction … All others bid less than me … Thus the object’s value must be lower than I thought • Winning the auction is “bad news” One must incorporate this into one’s bid Assume that your estimate is the most optimistic

  34. Avoiding the Winner’s Curse • Bidding for a company of uncertain value

  35. Avoiding the Winner’s Curse The expected value of the object is irrelevant. To bid: Consider only the value of the object if you win!

  36. Avoiding the Winner’s Curse • Bidding with no regrets: • Since winning means you have the most optimistic signal, always bid as if you have the highest signal • If your estimate is the most optimistic – what is the object worth? • Use that as the basis of your bid

  37. Summary • Average value of an object is irrelevant • Consider only the value if you win • In common value auctions, assume that you have the most optimistic estimate

  38. Extra Low Frequency (ELF) LF HF UHF EHF MF VHF SHF Infrared Visible Ultraviolet XRay Gamma Cosmic Ray Ray 3 x 10-8 m / 0 Hz 3 x 10-7 Å / 1025 Hz • “The greatest auction in history” - New York Times, March 16, 1995, p.A17

  39. More Bidders • More bidders lead to higher prices • Example • Second price auction • Each bidder has a valuation of either $20 or $40, each with equal probability • What is the expected revenue?

  40. Number of Bidders • Two bidders • Each has a value of 20 or 40 • There are four value combinations: Pr{20,20}=Pr{20,40}=Pr{40,20}=Pr{40,40}= ¼ Expected price = ¾ (20)+ ¼ (40) = 25

  41. Number of Bidders • Three bidders • Each has a value of 20 or 40 • There are eight value combinations: Pr{20,20,20}=Pr{20,20,40}=Pr{20,40,20} = Pr{20,40,40}=Pr{40,20,20}=Pr{40,20,40} = Pr{40,40,20}=Pr{40,40,40}= 1/8 Expected price = ½ (20)+ ½ (40) = 30

  42. Expected Price Number of Bidders Number of Bidders • Example: New Zealand 1993 UHF License Auction • Second price auction • Four lots won by Sky Network: • Assume more generally that valuations are drawn uniformly from [20,40]:

  43. Importance of RulesFCC Spectrum Auctions… • Want to encourage minority and female-owned firms to bid but licenses are very expensive. • Reserve several frequency blocks for smaller bidders. • Allow 10% down, low interest, remaining principal owed in 7 years. • What happens?

  44. “Tweaking the Rules” II(continued) • Bid high! • If licenses end up being worth less, default! • Of the four largest winners, • one went bankrupt and defaulted • one had $1B reduced to $66M in bankruptcy court • one was a front for Qualcomm • one was sold to Siemens

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